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Are the battery and wifi cable/airport card tested? Very likely liquid damage to airport card or the cable.
Does the battery indicator light up if you press the button without the battery installed?
Where was the original damage? Drop, liquid somewhere(where if liquid)? This is too weird a problem to just happen out of the blue. We need hints!
The last time I had a machine that would reboot when charger or battery was plugged in it was an SMC issue, but that was almost three years ago... and once, ever.
When you plug in a known good battery with a full charge, do you get all the green LEDs to light up on the battery indicator?